A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem

Abstract

The Multiple Knapsack Problem (MKP) (with equal capacities) can be defined as follows: Given a set of n items with positive integer weights and profits, a subset has to be selected such that the items in this subset can be packed into m knapsacks of equal capacities and such that the total profit of all items in the knapsacks is maximized. For m = 1 (MKP) reduces to the classical 0-1 single knapsack problem. It is known that (MKP) admits no fully polynomial-time approximation scheme even for m = 2 unless  $\mathcal{P} = \mathcal{NP}$. In this paper we present a polynomial time approximation scheme for (MKP) even if m is part of the input. This solves an important open problem in the field of knapsack problems.